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For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G\"odel's second incompleteness theorem that hold within second-order arithmetic.

For $n \in \mathbb{N}$ and $\varepsilon > 0$, given a sufficiently long sequence of events in a probability space all of measure at least $\varepsilon$, some $n$ of them will have a common intersection. A more subtle pattern: for any $0 < p < q < 1$, we cannot find events $A_i$ and $B_i$ so that $\mu \left( A_i \cap B_j \right) \leq p$ and $\mu \left( A_j \cap B_i\right) \geq q$ for all $1 < i < j < n$, assuming $n$ is sufficiently large. This is closely connected to model-theoretic stability of probability algebras. We survey some results from our recent work on more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory., Comment: 6 pages; to appear in the Proceedings of the 14th Panhellenic Logic Symposium

It is often convenient to represent a process for randomly generating a graph as a graphon. (More precisely, these give \emph{vertex exchangeable} processes -- those processes in which each vertex is treated the same way.) Other structures can be treated by generalizations like hypergraphons, permutatons, and, for a very general class, theons. These representations are not unique: different representations can lead to the same probability distribution on graphs. This naturally leads to questions (going back at least to Hoover's proof of the Aldous--Hoover Theorem on the existence of such representations) that ask when quasirandomness properties on the distribution guarantee the existence of particularly simple representations. We extend the usual theon representation by adding an additional datum of a random permutation to each tuple, which we call a $\ast$-representation. We show that if a process satisfies the \emph{unique coupling} property UCouple[$\ell$], which says roughly that all $\ell$-tuples of vertices ``look the same'', then the process is $\ast$-$\ell$-independent: there is a $\ast$-representation that does not make use of any random information about $\ell$-tuples (including tuples of length $<\ell$). Simple examples show that the use of $\ast$-representations is necessary. This resolves a question of Coregliano and Razborov, since it easily follows that UCouple[l] implies Independence[\ell'] (the existence of an $\ell'$-independent ordinary representation) for $\ell'<\ell$.

In this paper we give an ordinal analysis of the theory of second order arithmetic. We do this by working with proof trees -- that is, "deductions" which may not be well-founded. Working in a suitable theory, we are able to represent functions on proof trees as yet further proof trees satisfying a suitable analog of well-foundedness. Iterating this process allows us to represent higher order functions as well: since functions on proof trees are just proof trees themselves, these functions can easily be extended to act on proof trees which are themselves understood as functions. The corresponding system of ordinals parallels this, using higher order collapsing function., Comment: Work in progress

We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory., Comment: 67 pages